A293147 Triangle read by rows: coefficients of the characteristic polynomial of the n-th submatrix of A191898.

0, 1, -1, -2, 0, 1, 6, 4, -2, -1, 0, -12, -5, 3, 1, 0, 60, 49, -3, -7, -1, 0, 360, 84, -90, -19, 5, 1, 0, -2520, -1308, 414, 241, -5, -11, -1, 0, 0, 3780, 1752, -590, -290, 9, 12, 1, 0, 0, 0, -7560, -2874, 1122, 406, -19, -14, -1

Offset: 0

Author

Mats Granvik, Oct 01 2017

Keywords

• sign
• tabl

Comments

It appears that for n > 10, the nearest integer to the largest negative eigenvalue of the n-th characteristic polynomial is equal to the previous prime sequence A007917(n).

A007917(n) = round(max(-eigenvalues(A191898(1..n,1..n)))) (for n > 10), has been verified in the range n=11 to n=100.

Examples

			   0;
   1,    -1;
  -2,     0,     1;
   6,     4,    -2,    -1;
   0,   -12,    -5,     3,     1;
   0,    60,    49,    -3,    -7,   -1;
   0,   360,    84,   -90,   -19,    5,   1;
   0, -2520, -1308,   414,   241,   -5, -11,  -1;
   0,     0,  3780,  1752,  -590, -290,   9,  12,   1;
   0,     0,     0, -7560, -2874, 1122, 406, -19, -14, -1;
   ...
max(-eigenvalues(A191898(1..12,1..12)))=11.096...
max(-eigenvalues(A191898(1..13,1..13)))=12.9021...
		

Links

• Mats Granvik, Primes approximated by eigenvalues?
• Mats Granvik, Mathematica program to verify the agreement between the largest negative eigenvalues and the previous prime sequence.

Crossrefs

Cf. A191898, A007917.

Programs

Mathematica

Clear[A,B,nnn]; nnn=9; charpol = Table[A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; B = Table[Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {k, 1, nn}], {n, 1, nn}]; CoefficientList[CharacteristicPolynomial[A.B, x], x], {nn, 1, nnn}];Flatten[charpol]